<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html><head><title>Body Length</title>

<meta http-equiv="content-type" content="text/html; charset=utf-8" />
<meta http-equiv="content-style-type" content="text/css" />
<meta http-equiv="expires" content="" />
<meta name="lang" content="en-US" />
<meta name="template-info" content="NewTopicFile.htm 2010-03-04" />
<meta name="creation-date" scheme="iso8601" content="2014-05-09" />
<meta name="last-modified" scheme="iso8601" content="2014-05-22" />
<meta name="version" content="6" />
<meta name="author" content="Ulrich Sprick" />
<meta name="robots" content="follow" />
<meta name="category" content="Lutherie/Classical Guitar" />
<meta name="keywords" content="lutherie, guitar." />
<meta name="description" content="" />

<link rel="stylesheet" type="text/css" href="../../../includes/topic-2.css" />
<link rel="stylesheet" type="text/css" href="../includes/local.css" />
<link rel="stylesheet" type="text/css" href="../includes/calculator.css" />

<script language="JavaScript" type="text/javascript" src="../../../includes/global.js"></script>
<script language="JavaScript" type="text/javascript" src="../../../includes/cookies.js"></script>
<script language="JavaScript" type="text/javascript" src="../../../includes/topic.js"></script>
<script language="JavaScript" type="text/javascript" src="../includes/local.js"></script>
<script language="JavaScript" type="text/javascript" src="../includes/calculator.js"></script>
<script type="text/javascript" src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>

</head><body class="content" onload="onloadTopic()"><a name="top"></a>

<h1>Body Length</h1>

<p>This section develops the body length formula.</p>

<p class="imgbox"><a name="fig-059"></a><img src="../img/fig-059.png" style="max-width: 500px;"/><br />Figure 59: Calculating the body length.</p>

<h2>What We Know</h2>

<pre class="formula">
(1)	P0P1 = P0P2 = P0P3 = P0P6 = DR = dome radius
(2)	P1P5 = BP = bridge position, distance from top end of body
(3)	P5P6 = DH = dome height at bridge position
</pre>

<p>We know also that the dome height is measured down perpendicular to the reference line:</p>

<pre class="formula">
(4)	P5-P6 is perpendicular to P1-P2.
</pre>

<h2>What We Want</h2>

<pre class="formula">
(5)	P2P2 = BL = body length
</pre>

 <h2>Calculation Path</h2>
 
 <p>Step 1: We know that the angle at P5 in the triangle P1-P5-P6 is square, and we know the length of the sides <span class="overline">P1P5</span> and <span class="overline">P5P6</span> from (2) and (3). Then we can apply Pythagoras to calculate the length P1-P6:</p>

<pre class="formula">
(6)	P1P6<sup>2</sup> = P1P5<sup>2</sup> + P5P6<sup>2</sup>
(7)	P1P6 = sqrt( P1P5<sup>2</sup> + P5P6<sup>2</sup> ) 
</pre>

<p>Step2: Then we the create the perpendicular bisector of P1-P6 through P8:</p>

<pre class="formula">
(8)	P1P10 = P6P10 = P1P6 / 2
</pre>

<p class="imgbox"><a name="fig-054"></a><img src="../img/fig-054-2.png" style="max-width: 500px;"/><br />Figure 54</p>

<p>Because P0-P1 and P0-P6 are equal in length (equation (1)), the new helper line also goes through the dome origin P0, dividing the triangle P1-P0-P6 into two equal halves.</p>

<p>Step 3: Then we calculate <span class="overline">P0P10</span> with Pythagoras. </p>

<pre class="formula">
(9)	P0P10<sup>2</sup> = P0P6<sup>2</sup> - P6P10<sup>2</sup>
(10)	P0P10 = sqrt( P0P6<sup>2</sup> - P6P10<sup>2</sup> )
</pre>

<p>Step 4: Now that two sides are known (equations (1) and (10)), we can calculate the angle a1 in the triangle P1-P0-P8:</p>

<pre class="formula">
(11)	sin a1 = P1P10 / P0P1
(12)	a1 = arcsin( P1P10 / P0P1 )
</pre>

<p>Step 5: We drop a perpendicular on the reference line through P0. This is the familiar line that goes through the maximum dome height P3, and divides the body length P1-P1 into equal halves at P4. The intersection with P1-P6 will be named P9.</p>

<p class="imgbox"><a name="fig-055"></a><img src="../img/fig-055-2.png" style="max-width: 500px;"/><br />Figure 55</p>

<p>Step 7: In this step we calculate the angle a2. Because both lines P5-P6 and P0-P3 intersect the reference line P1-P2 at square angles, the angle a2 in P1-P6-P5 is the same as a2 in the triangle P1-P9-P4. So we can calculate a2 as</p>

<pre class="formula">
(13)	cos a2 = P5P6 / P1P6
(14)	a2 = arccos( P5P6 / P1P6 )
</pre>

<p>Step 8: Now that a2 is known, we can calculate the angle a3 in P8-P0-P9, because the sum of including angles in a triangle is always 180 degrees:</p>

<pre class="formula">
(15)	a3 = 90 - a2
</pre>

<p>Step 9: If we now take a look at the triangle P0-P4-P2, we find that the angle at P0 can easily found as</p>

<pre class="formula">
(16)	a4 = a1 + a3
</pre>

<p>Step 10: This means that the triangle P0-P4-P1 is now determined: We have the length of the hypothenusis P0-P1, and two angles, a4 at P0, and a right angle at P4. So we can calculate the position of P4:</p>

<pre class="formula">
(17)	sin a4 = P1P4 / P0P1
(18)	P1P4 = P0P1 * sin a4
</pre>

<p>Step 11: Because P0-P1 and P0-P2 have the same length (equation (1)), and P0-P3 intersects the P1-P2 at a right angle, P4 is in the middle of P1-P2:</p>

<pre class="formula">
(19)	P1P4 = P4P2 = P1P2 / 2
</pre>

<p>Step 12: Resolving for P1P2, we find the last step for calculating the body length:</p>

<pre class="formula intermediate">
(20)	P1P2 = 2 * P1P4
</pre>

<h2>Substitution of Intermediate Results</h2>

<p>Now we substitute the intermediate results in order to express equation 20 with known input values. </p>

<p>Introducing equation (18):</p>

<pre class="formula">
(21)	P1P2 = 2 * P0P1 * sin a4
</pre>

<p>With equation (16) we get:</p>

<pre class="formula">
(22)	P1P2 = 2 * P0P1 * sin (a1 + a3)
</pre>

<p>Angle a1 has been defined in equation (12), and a3 in (15):</p>

<pre class="formula">
(23)	P1P2 = 2 * P0P1 * sin ((arcsin( P1P10 / P0P1 )) + 90 - a2)
</pre>

<p>Angle a2 was found in (14):</p>

<pre class="formula">
(24)	P1P2 = 2 * P0P1 * sin ((arcsin( P1P10 / P0P1 )) + 90 - arccos( P5P6 / P1P6 ))
</pre>

<p>We found P1P10 in (8)</p>

<pre class="formula">
(25)	P1P2 = 2 * P0P1 * sin ((arcsin(P1P6 / 2 / P0P1 )) + 90 - arccos( P5P6 / P1P6 ))
</pre>

<p>P1-P6 was found in (7):</p>

<pre class="formula">
(26)	P1P2 = 2 * P0P1 * sin ((arcsin ((sqrt( P1P5<sup>2</sup> 
	+ P5P6<sup>2</sup> )) / 2 / P0P1 )) + 90
	- arccos( P5P6 / (sqrt( P1P5<sup>2</sup> + P5P6<sup>2</sup> ))))
</pre>

<p>With the definining equations (1)...(3) and (5) we have the final formula:</p>

<pre class="formula final">
(27)	BL = 2 * DR * sin ((arcsin ((sqrt( BP<sup>2</sup> + DH<sup>2</sup> )) / 2 / DR )) 
	+ 90 - arccos( DH / (sqrt( BP<sup>2</sup> + DH<sup>2</sup> ))))
</pre>


<!-- verified 2014-05-20  -->

<div class="footer"><hr />
<p>Page author: USP &bull; Page editor: USP &bull; Last update: 2014-05-22 &bull; 6</p>
<p>&copy; 2013 Synesis Ulrich Sprick. All rights reserved. See <a href="../../../copyright.htm">Copyright details</a>.</p>
<p><a href="#top">Back to Top</a> | <a ="../../../index.htm" target="_top">Home</a> | <a id="reload" onclick="reloadFrameset()" href ="../../index.htm" target="_top" >Reload Frameset</a></p>
</div></body></html>
